Clarification: As the Gemara explains - When a Sa'ah of T'rumah falls into a less than a hundred Sa'ah of Chulin and the became Meduma, and some of the Dimu'a then fell into another batch of Chulin, the second batch only becomes Meduma according to the amount of T'rumah that is now in it ...

Clarification (cont.): So that in the event that it contains sufficient to render Bateil the T'rumah that fell into it, it becomes Bateil, even though the first batch did not ...

Authorship: And this goes according to the Chachamim in the Gemara, and not like Rebbi Eliezer, who says that 'It is Medama like T'rumah Vadai'.

Rebbi Eliezer: And in the same way, Rebbi Eliezer's Din will apply even if the first batch did contain a hundred Sa'ah, enough to render Bateil the T'runmah that fell into it ...

Reason: Seeing as he did not remove one Sa'ah before it fell into the second batch ...

Rebbi Eliezer (cont.): And he will have to remove it ...

Reason): Due to Gezel ha'Sheivet.

Explanation #1: Rashi explains - a dough that rose via yeast of T'rumah is completely Asur; and if some of that dough then falls into another Chulin dough and causes it to rise, it only renders it Asur according to the measure of the T'rumah yeast that got mixed in it ...

Explanation #1 (cont.): In other words, the latter dough only becomes Asur if there was sufficient T'rumah yeast of the first dough to make it rise.

Explanation #1 (concl.): But if it requires the combination of all that what in the first dough to make it rise, then it is permitted.

Reason: Since not enough of the T'rumah fell to make it rise (until here is the wording of Rashi).

Refutation: This is not correct however, because if that is so, when the Gemara, with reference to that Beraisa, comments that it does not go according to Rebbi Eliezer, who says in a Mishnah 'A yeast of Chulin and T'rumah ... ', it did not to prove that it is not like Rebbi Eliezer from the Mishnah of 'Yeast' ...

Refutation (cont.): Since it could have extrapolated from the Mishnah that it already cited in connection with Meduma that it does not go like him?

Explanation #2: It therefore seems that it speaks where a k'Zayis of T'rumah yeast together with a k'Zayis of Chulin yeast both fell into a Chulin dough and caused it to rise, and neither of them would have done so on its own, that it is not Asur ...

Explanation #2 (cont.): And this is what is meant by 'according to the Cheshbon' - enough to make it rise via the T'rumah on its own ...

Conclusion: And that conforms precisely to the Mishnah cited in the Gemara 'And the Chachamim say ... ', since he (Rebbi Eliezer) holds 'Zeh v'Zeh Gorem Asur'.

Clarification: Because She'ivah (drawn water) is Metaher via a majority and Hamshachah (flowing along the ground) ...

Clarification (cont.): And that is 'according to the Cheshbon' - in that She'uvin only disqualifies the Mikvah if the amount equals the Kasher water - twenty Sa'ah of She'uvin and twenty Sa'ah of Kasher water ...

Clarification (cont.): But if he was only Moshech nineteen Sa'ah of She'uvin into twenty-one of Kasher water, the Mikvah is Kasher.

Explanation #1: This means that it fell from three vessels corresponding to the number of Lugin - and the author is Yosef ben Choni.

Explanation #1 (cont.): According to that, the Rabbanan who argue are more lenient, and the case of 'When Ravin came' goes according to the Rabbanan ...

Explanation #1 (concl.): Whereas our Mishnah is not speaking about that, that we should say that it is Rebbi Eliezer and the Rabbanan are more stringent (Rashi).

Question #1: ha'Rav Shmuel from Lunil queries this however, in that it is not the way of the Gemara at all - to say that initially, according to Rebbi Chiya bar Aba, Rebbi Eliezer is lenient and the Rabbanan strict, and then according to Ravin, Rebbi Eliezer is strict and the Rabbanan lenient - such a reversal of opinions among the Amora'im we do not find? (See Shitah Mekubetzes 30).

Introduction to Question #2: Furthermore, how can one say that, according to the Rabbanan, drawn water which, via Hamshachah, makes up the entire Mikvah, which contains no rainwater, is Kasher, since ...

Introduction to Question #2 (cont.): This poses a Kashya from the first Perek of Ta'anis (Daf 13a) where, on the Mishnah 'One closes the bathhouses, the Gemara comments 'Whatever is due to mourning, it is forbidden to bathe in both hot and cold water; whereas whatever is merely for pleasure is forbidden in hot water but permitted in cold ...

Introduction to Question #2 (concl.): Till 'Is this a proof for him - "All Chayvei Tevilos may Tovel as normal on Tish'ah be'Av and on Yom Kipur". How is this speaking? If it is speaking about hot water, how can one Tovel in hot water, which is She'uvin?'

Question #2: According to Rashi, that drawn water which, via Hamshachah, makes up the entire Mikvah, which contains no Kasher water, is Kasher without any rain-water, we can indeed find Tevilah in hot water - where one made Hamshachah?

Introduction to Question #3: There is another Kashya from the Mishnah in Mikva'os (Perek 6 Mishnah 8) - 'If there are two Mikvaos, one higher up and one lower down, the former contains forty Sa'ah, but the latter does not - then he may fill buckets manually and pour them into the upper Mikvah so that it overflows into the lower one until it contains forty Sa'ah' ...

Question #3: Now if She'ivah was Metaher even without any rainwater, why would one need to pour into the upper Mikvah? It ought to be Tahor even if he pours directly on to the ground and it flows into the lower Mikvah itself?

Introduction to Question #4: Moreover, the Mishnah says there in Mikva'os (Mishnah 4) 'If rainwater and Mayim Shye'uvin mix in a Chatzer in a pit or on the steps of a cave, if the majority is the Kasher water, it is Kasher, but if the Pasul water is the majority, it is Pasul; half half, it is also Pasul ...

Introduction to Question #4 (cont.): When is that? Where they did not mix before reaching the Mikvah ... ' ...

Question #4: Why do they need to mix at all, seeing as even drawn water on its own is Kasher via Hamshachah?

Explanation #2: ha'Rav Shmuel therefore sent that he had received from his Rebbes that 'Mayim She'uvim which is completely drawn is Tahor' speaks where forty Sa'ah of rainwater subsequently came into the Mikvah (See Shitah Mekubetzes Hashmatos).

Explanation #2 (cont.): According to that, all the Kashyos are answered and the statement of Ravin can very well go like Rebbi Eliezer ben Ya'akov - because with regard to combining to form forty Sa'ah, one needs a majority of Kasher water.

Support: And this is also implied in the Lashon of the She'iltos of Rav Acha'I, where it writes 'A Mikvah into which three Lugin of drawn water fall is Pasul; but if one makes Hamshachah from outside via a drain and it then flows into the Mikvah, it does not render it Pasul ...

Support (cont.): Because when Ravin came he quoted Rebbi Yochanan as saying that 'Drawn water which, via Hamshachah, makes up the entire Mikvah is Tahor, and it combines to make up forty Sa'ah - as we learned in a Beraisa 'Rebbi Eliezer ben Ya'akov says ... ' ...

Support (concl.): So we see that he holds that one can establish both the ruling of Rebbi Eliezer ben Ya'akov and that of Ravin, and they do not contradict one another.

Refutation: The Ri however, answered him that one can establish our Sugya ... since it seems to him that Rebbi Eliezer ben Ya'akov requires a majority of rainwater, but if the majority is drawn water, even if forty Sa'ah of rainwater subsequently falls into it, iyt does not become Kasher ...

Refutation (cont.): And that explains why the Gemara asks that the statement of Ravin goes neither like Rebbi Eliezer ben Ya'akov nor like the Rabbanan.

Refuting R. Shmuel's Questions: However the Ri says that all that he (R. Shmuel) asked is not difficult ...

Refuting Question #1: Because in many places in Shas we find that initially it explains the words of one Tana to be stringent, and then it explains it to be lenient.

Example #1: In the first Perek of Kidushin (Daf 37a) in connection with Rebbi Eliezer, where the Gemara asks whether Rebbi Eliezer argues le'Kula or le'Chumra.

Examples #2 & #3: And similarly in connection with a Mavoy whose Lechi was removed, and in connection with Chelmi.

Refuting Question #2: Nor is the Sugya in Ta'anis difficult - since even according to the questioner one can also ask that one does find Tevilah in hot water - there where one is Mochech nineteen Sa'ah of hot water into twenty-one of rainwater ...

Refuting Question #2 (cont.): Or where one poured any amount into a complete Mikvah - because once it is complete, it does not become Pasul

Refuting Question #2 (cont.): Or via metal bars that were heated and placed in the Mikvah, as the Gemara states in Yoma (Daf 31b) in connection with the Tevilah of the Kohen Gadol on Yom Kipur.

Refuting Question #2 (concl.): And it is possible to establish the remaining Mishnyos that Tosfos quoted from Masechtes Mikva'os like Rebbi Eliezer ben Ya'akov.

Refuting Question #3: Moreover one can answer that the Mishnah 'He places the water in the upper Mikvah until forty Sa'ah overflows into the lower Mikvah' - is coming to teach us that once a Mikvah contains forty Sa'ah, even if a hundred Sa'ah of Mayim She'uvin fall into it, they do not render it Pasul.

Halachah: It would seem that according to the Halachah one may not make a Mikvah entirely of Mayim She'uvim via Hamshachah ...

Reason: Since even according to Rashi, the Halachah is like Rebbi Eliezer ben Ya'akov, whose Mishnah is 'Kav ve'Naki' (sparse but clean - i.e. Halachah) ...

Halachah (cont.): But via a majority and Hamshachah it is unanimously Kasher ...

Clarification: Inasmuch as if the Mikvah contains twenty-one Sa'ah of Kasher water, it is permitted to fill it manually with nineteen Sa'ah, to make a ditch in the ground and pour the water into it and let it flow into the Mikvah until it fills up ...

Source: Like the Mishnah in the fourth Perek of Mikva'os wich Tosfos quoted - 'When is that? When they get mixed before they reach the Mikvah' - so it concludes that via a majority and Hamshachah everyone agrees that it is Kasher.

Implied Question: And the reason that the Tana says 'before they reach the Mikvah' is ...

Answer: To preclude where the pipe that they carved and then fitted goes right down to the Mikvah without Hamshachah (flowing along the ground).

Clarification: And 'L'fi Cheshbon, according to this, means - that the drawn water does not disqualify the Mikvah unless it falls into it in accordance with the number of Lugin that render the Mikvah Pasul - i.e. three, but if the three Lugin fall in four or five times, they do not disqualify it (See Shitah Mekubetzes Hashmatos).

P'sak Halachah: ha'Rav Avraham b'R. David Paskens like Yosef ben Choni ...

Inference: It would therefore seem that when one drains the Mikvah to clean it, taking out the water and replacing it, one need not worry if some of the original water remains in the Mikvah ...

Implied Question: Even if some of it falls from the bucket into it, and that more than three Lugin remain ...

Answer: Nevertheless, it doesn't matter, since the three Lugin did not fall in from three vessels.

Another Opinion: However, some people are Machmir - they make a hole in the bucket large enough to allow liquid to enter, due to the concern that Tosfos just raised.

Question: In that case, what do we learn from the Gezeirah-Shavah?

Answer: It is a Mitzvah to pour the water first and then the earth, as the Torah writes in connection with a Sotah (Bamidbar 5) "And from the earth that is on the ground of the Mishkan the Kohen shall take and put it into the water" ...

Conclusion: And that is what the Gezeirah Shavah comes to teach us.